Numerical Simulation of Dynamic Systems XXVII · Numerical Simulation of Dynamic Systems XXVII...

Numerical Simulation of Dynamic Systems XXVII


Prof. Dr. Francois E. CellierDepartment of Computer Science

ETH Zurich

May 28, 2013


Logarithmic Quantization

Introduction

Introduction

This 27th and final presentation of the class on the Numerical Simulation of DynamicSystems deals with “Chapter 13 of a 12-chapter book,” i.e., it discusses a number ofissues related to Quantized State System (QSS) solvers that were discovered onlyafter the book went to print.



Introduction

Introduction


� Until now, we have always controlled the absolute integration error. It may bemore robust to control the relative integration error, as this quantity doesn’tneed to be set for each simulation separately. Hence being able to specify therelative error tolerance makes the software more robust and user-friendly.



Introduction

Introduction



� Until now, we have only dealt with non-stiff QSS-based integrators. Yet, weknow from the earlier presentations that large systems are almost invariably stiff,and consequently, we shall need also stiff QSS-based integrators. Suchalgorithms have meanwhile been developed and shall be discussed in thispresentation.



Introduction

Introduction



� Until now, we have only dealt with non-stiff QSS-based integrators. Yet, weknow from the earlier presentations that large systems are almost invariably stiff,and consequently, we shall need also stiff QSS-based integrators. Suchalgorithms have meanwhile been developed and shall be discussed in thispresentation.

� Also of importance are special solvers for dealing with marginally stable systems,and indeed, an F-stable QSS-based solver has also been developed since thebook went to press.



Introduction

Introduction II

� A type of models that we haven’t dealt with in this class is models that containdelays. We call a set of ordinary differential equations with delays a delaydifferential equation (DDE) model, and we call algorithms that are suitable forthe simulation of DDE models numerical DDE solvers. A QSS-based DDEsolver has meanwhile also been developed and shall be discussed.



Introduction

Introduction II

� A type of models that we haven’t dealt with in this class is models that containdelays. We call a set of ordinary differential equations with delays a delaydifferential equation (DDE) model, and we call algorithms that are suitable forthe simulation of DDE models numerical DDE solvers. A QSS-based DDEsolver has meanwhile also been developed and shall be discussed.

� Finally, a few remarks shall be offered on the use of QSS-based algorithms inreal-time simulations.




Choosing the Quantum Size

Let us suppose that we wish to simulate a system in which a state variable x1 growsfrom 0 to 1000, while x2 grows from 0 to 1.






� Using QSS, if we use a quantum of ΔQ1 = ΔQ2 = 0.1, variable q1 will change10, 000 times, while variable q2 will change only 10 times.







� Using QSS2 or QSS3, the number of steps differs in a less dramatic way, but itstill depends on the amplitude of the signals to be integrated.








� Consequently, an appropriate value of the quantum used for each state variabledepends on the variability of that state variable.









� Unfortunately, before performing the simulation, we normally do not know, howthe state variables will develop over time.









� Unfortunately, before performing the simulation, we normally do not know, howthe state variables will develop over time.

We need some mechanism to adjust the size of the quantum automatically duringthe simulation in accordance with the values that the state variable assumes overtime.





We shall set the quantum proportional to the absolute value of each state. Also, toavoid problems around zero, we shall limit the minimum quantum value.






Thus, we shall express the quantum size as:

ΔQi (t) = max(ΔQreli · |xi (tk )|, ΔQmini)

where tk is the time of the last change of qi (t).









The parameters Qrel and Qmin can be specified for each hysteretic quantizedintegrator separately.









The parameters Qrel and Qmin can be specified for each hysteretic quantizedintegrator separately.

This idea can be applied to all QSS methods.




Logarithmic Quantization II

Logarithmic Quantizer

ΔQmin

x(t)

q(t)



Relative Error Control

Perturbed Representation

Recall that the QSS approximation of a linear time-invariant ODE:

xa(t) = A · xa(t) + B · u(t)

can be written as:x(t) = A · q(t) + B · u(t)






xa(t) = A · xa(t) + B · u(t)


Defining Δx(t) � q(t) − x(t), we have:

x(t) = A · (x(t) + Δx(t)) + B · u(t)






xa(t) = A · xa(t) + B · u(t)


Defining Δx(t) � q(t) − x(t), we have:

x(t) = A · (x(t) + Δx(t)) + B · u(t)

This perturbed representation does not depend on the type of quantization. However,with logarithmic quantization, the upper bound of |Δx(t))| depends on the state valuex(t).




Stability and Error Bound

Assume that matrix A = V · Λ · V−1 is Hurwitz, and define:

R � |V| · |Re(Λ)−1Λ| · |V−1|




Stability and Error Bound

Assume that matrix A = V · Λ · V−1 is Hurwitz, and define:

R � |V| · |Re(Λ)−1Λ| · |V−1|

Let ΔQrel be a diagonal matrix with entries ΔQreli , and let ΔQmin be a column vectorwith components ΔQmini

. Then, provided that all eigenvalues of the matrix R · ΔQrel

lie inside the unit circle, it can be shown that:

|e(t)| ≤ (I(n) − R · ΔQrel)−1 · R · max(ΔQrel · xmax,ΔQmin)

where xmax is the column vector of the maximum absolute values reached by eachcomponent of the analytical solution xa(t).




Logarithmic Quantization and Relative Error Control

� Generally, ΔQrel is chosen small, which ensures that all eigenvalues of R · ΔQrel

lie inside the unit circle.







� In this case, we can approximate (I(n) − R · ΔQrel)−1 ≈ I(n).








� Also, xmax is normally much bigger than ΔQmin.









� Usually, we shall choose ΔQreli = tolrel for all i .










In this case, we obtain:

|e(t)| ≤ (I(n) − R · ΔQrel)−1 · R · max(ΔQrel · xmax,ΔQmin) ≈ R · tolrel · |xmax|










In this case, we obtain:

|e(t)| ≤ (I(n) − R · ΔQrel)−1 · R · max(ΔQrel · xmax,ΔQmin) ≈ R · tolrel · |xmax|

The use of logarithmic quantization yields an intrinsic control of the relative error.


QSS Methods for Stiff Systems

QSS Methods and Stiff Systems


The linear time-invariant system:

x1(t) = 0.01 x2(t)

x2(t) = −100 x1(t) − 100 x2(t) + 2020

has eigenvalues λ1 ≈ −0.01 and λ2 ≈ −99.99, and consequently, the model is stiff.






x1(t) = 0.01 x2(t)

x2(t) = −100 x1(t) − 100 x2(t) + 2020


The QSS method approximates this system by:

x1(t) = 0.01 q2(t)

x2(t) = −100 q1(t) − 100 q2(t) + 2020






x1(t) = 0.01 x2(t)

x2(t) = −100 x1(t) − 100 x2(t) + 2020


The QSS method approximates this system by:

x1(t) = 0.01 q2(t)

x2(t) = −100 q1(t) − 100 q2(t) + 2020

Let us simulate this system using the QSS solver with initial conditions x1(0) = 0 andx2(0) = 20 and with the quantization ΔQ1 = ΔQ2 = 1.




QSS Methods and Stiff Systems II

0 50 100 150 200 250 300 350 400 450 500−5

0

5

10

15

20

25

t

q1(t

),q

2(t

)

q1(t)

q2(t)




QSS Methods and Stiff Systems III

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6−5

0

5

10

15

20

25

t

q1(t

),q

2(t

)

q1(t)

q2(t)




QSS Methods and Stiff Systems IV

� Stiff systems usually provoke fast oscillations on the QSS solutions.






� Consequently, the number of steps is very large.







� In the simulated example, there were 21 changes in q1 and 15, 995 changes inq2 for a final simulation time of tf = 500 seconds .







� In the simulated example, there were 21 changes in q1 and 15, 995 changes inq2 for a final simulation time of tf = 500 seconds .

Evidently, the QSS method is not appropriate for the simulation of stiff systems.



Backward QSS Method

Backward QSS: Basic Idea

How can we prevent oscillations?



Backward QSS Method



� In classical solvers, we had to use future values of state derivatives, i.e., implicitmethods, in order to obtain stiff system solvers.



Backward QSS Method




� Let us try to do the same also in QSS-based methods. We shall choose qj suchthat it represents a future value of xj .



Backward QSS Method





� In QSS, after a step in qj , we set qj (t) = xj(t). Now, we shall setqj (t) = xj (t) ± ΔQj depending on the sign of xj (t).



Backward QSS Method






� It may happen that xj(t)∣∣xj +ΔQj

< 0 and xj (t)∣∣xj−ΔQj

> 0. In that case, we

cannot choose a suitable value for qj . However in that case, there must exist avalue qj ∈ (xj − ΔQj , xj + ΔQj ) for which xj = 0.



Backward QSS Method










� If this happens, we keep qj (t) = xj(t), but set xj(t) = 0, i.e., no further internaltransition gets scheduled for the state variable in question.



Backward QSS Method










� If this happens, we keep qj (t) = xj(t), but set xj(t) = 0, i.e., no further internaltransition gets scheduled for the state variable in question.

This is an implicit algorithm, but it does not require a Newton iteration, as qj (t) canonly assume one of two values at each step. The method is called Backward QSS(BQSS) solver.



Backward QSS Method

Backward Quantization

Backward Quantizer

ΔQ

Input

Output



Backward QSS Method

Backward Quantization II

� In non-stiff QSS solvers, the trajectories of q(t) and x(t) at the beginning of thestep always move away from each other. The next internal transition isscheduled to occur at time t, where |q(t−) − x(t)| = ΔQ. At that time, q(t) isreset to q(t+) = x(t).



Backward QSS Method



� In stiff QSS solvers, the trajectories of q(t) and x(t) at the beginning of thestep always move toward each other. The next internal transition is scheduledto occur at time t, where |q(t−) − x(t)| = 0. At that time, q(t) is reset toq(t+) = x(t) ± ΔQ.



Backward QSS Method




� In BQSS, if q(t+) cannot be chosen such that the trajectories move towardeach other, then no internal transition is scheduled, i.e., the time advancefunction is set to ∞, and x(t) is set to x(t) = 0 irrespective of the current valueof the input signal to the hysteretic quantized integrator.



Backward QSS Method




� In BQSS, if q(t+) cannot be chosen such that the trajectories move towardeach other, then no internal transition is scheduled, i.e., the time advancefunction is set to ∞, and x(t) is set to x(t) = 0 irrespective of the current valueof the input signal to the hysteretic quantized integrator.

� Higher-order stiff QSS solvers need additional provisions that shall be discussedin due course.



Backward QSS Method

Backward QSS: An Example

1

x1, q1

05 10

x1 = 0.21

q1 = 1

x1 = 0.01q2

t = 4.762

21

x2, q2

20t

t

x2 = 20

x2 = −100q1 − 100q2 + 2020

q2 = 21

19

18

2



Backward QSS Method

Backward QSS: An Example

1

x1, q1

0

x1 = 0.19

q1 = 2

x1 = 0.01q2

t = 4.774

21

x2, q2

20t

t

x2 = −80

x2 = −100q1 − 100q2 + 2020

q2 = 1919

18

2

t = 4.762



Backward QSS Method

Backward QSS: An Example II

0 50 100 150 200 250 300 350 400 450 500−5

0

5

10

15

20

25

t

q1(t

),x 1

(t),

q2(t

),x 2

(t)

x1(t), q1(t)

x2(t), q2(t)



Backward QSS Method

Quantization Functions in BQSS

xi

qi

ΔQi

ε



Backward QSS Method


xi

qi

ΔQi

ε

� BQSS uses a lower andan upper hystereticquantization functionfor each state.



Backward QSS Method


xi

qi

ΔQi

ε


� They compute a lowerand an upper quantizedstate, q

jand qj .



Backward QSS Method


xi

qi

ΔQi

ε


� They compute a lowerand an upper quantizedstate, q

jand qj .

� The quantized state qj

is chosen from them inaccordance with thesign of xj .



Backward QSS Method

BQSS Definition

Given the system:x(t) = f(x(t), u(t))



Backward QSS Method

BQSS Definition


BQSS approximates it as:x(t) = f(q(t), u(t)) + Δf

where the components qj of q take the values of either the lower quantized state qj(t)

or the upper quantized state qj (t), such that:

fj(q(t), u(t)) · (qj (t) − xj (t)) > 0



Backward QSS Method

BQSS Definition


BQSS approximates it as:x(t) = f(q(t), u(t)) + Δf

where the components qj of q take the values of either the lower quantized state qj(t)

or the upper quantized state qj (t), such that:

fj(q(t), u(t)) · (qj (t) − xj (t)) > 0

∨

∃qj ∈ (qj(t), qj (t))

∣∣fj (q

(j)(t),u(t))= 0

with q(j)(t) = [q1, . . . , qj , . . . , qn]T .



Backward QSS Method

BQSS Definition II

In the BQSS approximation:

x(t) = f(q(t), u(t)) + Δf

the terms Δfj will satisfy:

Δfj =

{0, if fj (q(t), u(t)) · (qj − xj) > 0−fj(q(t), u(t)), otherwise



Backward QSS Method

Algorithm to Obtain q(t)

When a state xi (t) reaches the quantized state qi (t), we proceed as follows:



Backward QSS Method

Algorithm to Obtain q(t)

When a state xi (t) reaches the quantized state qi (t), we proceed as follows:

Algorithm:

1. We update qiand qi and choose qi = qi or qi = q

idepending on the sign of

xi (t−).

2. We evaluate the functions fj depending on qi .

3. If some fj changes its sign:

� We change qj according to the new sign of xj .

Otherwise

� End of the Algorithm.

4. We evaluate the functions fj depending on the quantized states qi that changedand we return to step 3.

Restriction: We do not allow any qj to change more than once in the process.



Backward QSS Method

BQSS and PowerDEVS

� The block diagram structure of BQSS is the same as that of QSS.

q1(t)

qn(t)

.

.

.

u(t)

q(t)

d1(t)

dn(t)

f1(·)

fn(·) Int. BQSS

Int. BQSS



Backward QSS Method

BQSS and PowerDEVS


q1(t)

qn(t)

.

.

.

u(t)

q(t)

d1(t)

dn(t)

f1(·)

fn(·) Int. BQSS

Int. BQSS

� The static functions are the same as for QSS.



Backward QSS Method

BQSS and PowerDEVS


q1(t)

qn(t)

.

.

.

u(t)

q(t)

d1(t)

dn(t)

f1(·)

fn(·) Int. BQSS

Int. BQSS

� The static functions are the same as for QSS.

� The hysteretic quantized integrators are similar, but:

� they compute qi in accordance with the algorithm just outlined,

� when the calculated value of qi does not agree with the sign of xi , they

act as if xi = 0.



Backward QSS Method

Stability and Error Bound of BQSS

BQSS is very similar to QSS, but it has an additional perturbation term that enforcesthat xi = 0 when a consistent value for qi cannot be found.

x(t) = f[q(t), t] + Δf(t) = f[x(t) + Δx(t), t] + Δf(t)

where |Δxi | ≤ ΔQi + εi .



Backward QSS Method





As a consequence:



Backward QSS Method





As a consequence:

� The global error formula of BQSS for linear time-invariant stable systems nowbecomes:

|xa(t) − x(t)| ≤ |V| · |Re(Λ)−1V−1| · |A| · (ΔQ + ε) ; ∀t ≥ 0



Backward QSS Method





As a consequence:



� The hysteresis width ε should now be chosen smaller than the quantum, so itdoes not introduce unnecessary errors.



Backward QSS Method





As a consequence:



� The hysteresis width ε should now be chosen smaller than the quantum, so itdoes not introduce unnecessary errors.

� The simulation trajectories are numerically stable for any quantum adopted.



Backward QSS Method

Example: Second-order Linear System

� Eigenvalues

λ1 = −0.01;λ2 = −99.99



Backward QSS Method


� Eigenvalues

λ1 = −0.01;λ2 = −99.99

� Initial states

x1(0) = 0; x2(0) = 20



Backward QSS Method


� Eigenvalues

λ1 = −0.01;λ2 = −99.99

� Initial states

x1(0) = 0; x2(0) = 20

� QuantizationS1 ΔQ1,2 = 1S2 ΔQ1,2 = 0.1S3 ΔQ1,2 = 0.01



Backward QSS Method


� Eigenvalues

λ1 = −0.01;λ2 = −99.99

� Initial states

x1(0) = 0; x2(0) = 20


� Number of EventsS1 x1 : 20; x2 : 22S2 x1 : 201; x2 : 201S3 x1 : 2006; x2 : 2024



Backward QSS Method


� Eigenvalues

λ1 = −0.01;λ2 = −99.99

� Initial states

x1(0) = 0; x2(0) = 20



� Error BoundS1 e1 ≤ 3; e2 ≤ 5S2 e1 ≤ 0.3; e2 ≤ 0.5S3 e1 ≤ 0.03; e2 ≤ 0.05



Backward QSS Method


� Eigenvalues

λ1 = −0.01;λ2 = −99.99

� Initial states

x1(0) = 0; x2(0) = 20




x1(t) = 0.01 x2(t)

x2(t) = −100 x1(t) − 100 x2(t) + 2020



Backward QSS Method


� Eigenvalues

λ1 = −0.01;λ2 = −99.99

� Initial states

x1(0) = 0; x2(0) = 20




x1(t) = 0.01 x2(t)

x2(t) = −100 x1(t) − 100 x2(t) + 2020

0 50 100 150 200 250 300 350 400 450 500−5

0

5

10

15

20

25

Time

qi

ΔQi = 1ΔQi = 0.1



Backward QSS Method

Example: Second-order Linear System II

Some remarks:



Backward QSS Method


Some remarks:

� As there are no oscillations, the number of events is the division between eachsignal amplitude and the corresponding quantum.



Backward QSS Method


Some remarks:


� Consequently, the number of steps grows inversely proportional to thequantization.



Backward QSS Method


Some remarks:



� The simulation reaches a stable situation some time before t = 500 seconds ,and after that, the simulation stops, as no further events are being scheduled.



Backward QSS Method


Some remarks:




� The theoretical error bound is conservative in this example.



Backward QSS Method

Main Features of BQSS

Advantages:



Backward QSS Method


Advantages:

� BQSS solves the problem of high-frequency oscillations exhibited whensimulating stiff systems using explicit QSS methods.



Backward QSS Method


Advantages:


� BQSS does not require Newton iterations, which permits the efficient numericalsimulation of many stiff systems.



Backward QSS Method


Advantages:



� BQSS is also efficient for discontinuity handling.



Backward QSS Method


Advantages:




� BQSS can be used for real-time simulation of stiff systems.



Backward QSS Method


Advantages:





Disadvantages:



Backward QSS Method


Advantages:





Disadvantages:

� Spurious equilibrium points may appear in nonlinear systems due to the termsΔfi .



Backward QSS Method


Advantages:





Disadvantages:

� Spurious equilibrium points may appear in nonlinear systems due to the termsΔfi .

� The number of steps grows linearly with the accuracy. Unfortunately, we havenot found any way to extend the BQSS algorithm to higher orders ofapproximation accuracy.



Linearly Implicit QSS Methods

LIQSS: Basic Idea

The Linearly Implicit QSS (LIQSS) algorithm is based on an idea very similar to thatof BQSS.




LIQSS: Basic Idea


� Just as in BQSS, we shall choose qj such that it represents a future value of xj .




LIQSS: Basic Idea



� Just as in BQSS, we set qj (t) = xj(t) ± ΔQj depending on the sign of xj(t).




LIQSS: Basic Idea




� The main difference with BQSS is that, when we cannot find qj such that xj

moves towards it, we now attempt to determine the value of qj that makesxj = 0, instead of enforcing xj = 0 by adding a perturbation term Δfj .




LIQSS: Basic Idea






� The search for qj is done in a linear manner.




LIQSS: Basic Idea






� The search for qj is done in a linear manner.

� Contrary to BQSS, once we have computed a new value for qj , no furtherinternal transition will be scheduled, until xj has advanced by ±ΔQj , even if thesign of xj changes during the step.




LIQSS and PowerDEVS

� The block diagram structure of LIQSS is the same as that of QSS and BQSS.

q1(t)

qn(t)

.

.

.

u(t)

q(t)

d1(t)

dn(t)

f1(·)

fn(·) Int. LIQSS

Int. LIQSS




LIQSS and PowerDEVS


q1(t)

qn(t)

.

.

.

u(t)

q(t)

d1(t)

dn(t)

f1(·)

fn(·) Int. LIQSS

Int. LIQSS

� The static functions are the same as for QSS and BQSS.




LIQSS and PowerDEVS


q1(t)

qn(t)

.

.

.

u(t)

q(t)

d1(t)

dn(t)

f1(·)

fn(·) Int. LIQSS

Int. LIQSS

� The static functions are the same as for QSS and BQSS.


� before they output a new value for qi , they estimate the value of the

derivative xi that each of the two options would produce;

� if one of them has the correct sign, i.e., xi → qi , that value is chosen;

� otherwise, the value qi is estimated that makes xi = 0.




Stability and Error Bound of LIQSS







Its LIQSS approximation is governed by the same expression as that of QSS:

x(t) = f(q(t), u(t)) = f(x(t) + Δx(t), u(t))








However, qi can now differ from xi by an amount of 2 · ΔQi (after sign changes in xi ).








However, qi can now differ from xi by an amount of 2 · ΔQi (after sign changes in xi ).

The properties of LIQSS are almost identical to those of QSS, but the error boundis twice the error bound of QSS.





� Eigenvalues

λ1 = −0.01;λ2 = −99.99





� Eigenvalues

λ1 = −0.01;λ2 = −99.99

� Initial states

x1(0) = 0; x2(0) = 20





� Eigenvalues

λ1 = −0.01;λ2 = −99.99

� Initial states

x1(0) = 0; x2(0) = 20




x1(t) = 0.01 x2(t)

x2(t) = −100 x1(t) − 100 x2(t) + 2020





� Eigenvalues

λ1 = −0.01;λ2 = −99.99

� Initial states

x1(0) = 0; x2(0) = 20




x1(t) = 0.01 x2(t)

x2(t) = −100 x1(t) − 100 x2(t) + 2020

0 50 100 150 200 250 300 350 400 450 500−5

0

5

10

15

20

25

Time

qi

ΔQi = 1ΔQi = 0.1





Some remarks:




Main Features of LIQSS

Advantages:





Advantages:

� LIQSS shares the main advantages of BQSS.





Advantages:


� LIQSS solves the problem of spurious equilibrium points of BQSS.





Advantages:



Disadvantages:





Advantages:



Disadvantages:

� The number of steps still grows linearly with the accuracy. However, it ispossible to obtain higher-order LIQSS methods.





Advantages:



Disadvantages:

� The number of steps still grows linearly with the accuracy. However, it ispossible to obtain higher-order LIQSS methods.

� LIQSS only looks at the main diagonal of the Jacobian matrix, allowing itsometimes to happen that xj changes its direction without causing amodification of qj . In such cases, LIQSS can provoke oscillations depending onthe system structure, i.e., the algorithm may lose its stiffly-stable nature.




LIQSS2: Basic Idea

� LIQSS2 is similar to QSS2. The quantized variables qj (t) are piecewise linear.




LIQSS2: Basic Idea


� Each section of qj (t) starts at xj(t) ± ΔQj with a slope of qj (t) = xj(t)|qj (t).




LIQSS2: Basic Idea



� We shall attempt to ensure, just as in the case of LIQSS, that xj (t) movestoward qj (t). That is, we demand that xj(t) · (qj (t) − xj (t)) > 0.




LIQSS2: Basic Idea




� When this is not possible, we shall use a linear approximation to find the valueqj (t) with slope qj (t) = xj (t)|qj (t), for which xj(t) = 0.




LIQSS2: Basic Idea




� When this is not possible, we shall use a linear approximation to find the valueqj (t) with slope qj (t) = xj (t)|qj (t), for which xj(t) = 0.

� Although this is an implicit algorithm, there is no need for Newton iterations totake place, as only three possible trajectories can be chosen for qj (t).




LIQSS2: Trajectories




LIQSS2 and PowerDEVS

� The block diagram structure of LIQSS2 is the same as that of QSS2.

q1(t)

qn(t)

.

.

.

u(t)

q(t)

d1(t)

dn(t)

f1(·)

fn(·) Int. LIQSS2

Int. LIQSS2






q1(t)

qn(t)

.

.

.

u(t)

q(t)

d1(t)

dn(t)

f1(·)

fn(·) Int. LIQSS2

Int. LIQSS2

� The static functions are the same as for QSS2.






q1(t)

qn(t)

.

.

.

u(t)

q(t)

d1(t)

dn(t)

f1(·)

fn(·) Int. LIQSS2

Int. LIQSS2

� The static functions are the same as for QSS2.


� before they output a new value for qi , they estimate the value of the

second derivative xi that each of the two options would produce;

� if one of them has the correct sign, i.e., xi → qi , that value is chosen;

� otherwise, the value qi is estimated that makes xi = 0.




Stability and Error Bound of LIQSS2







Its LIQSS2 approximation is governed by the same expression as that of QSS:









However, just as in the case of LIQSS, qi can now differ from xi by an amount of2 · ΔQi (after sign changes in xi ).








However, just as in the case of LIQSS, qi can now differ from xi by an amount of2 · ΔQi (after sign changes in xi ).

The properties of LIQSS2 are identical to those of LIQSS.





� Eigenvalues

λ1 = −0.01;λ2 = −99.99





� Eigenvalues

λ1 = −0.01;λ2 = −99.99

� Initial states

x1(0) = 0; x2(0) = 20





� Eigenvalues

λ1 = −0.01;λ2 = −99.99

� Initial states

x1(0) = 0; x2(0) = 20




x1(t) = 0.01 x2(t)

x2(t) = −100 x1(t) − 100 x2(t) + 2020





� Eigenvalues

λ1 = −0.01;λ2 = −99.99

� Initial states

x1(0) = 0; x2(0) = 20




x1(t) = 0.01 x2(t)

x2(t) = −100 x1(t) − 100 x2(t) + 2020

0 50 100 150 200 250 300 350 400 450 5000

5

10

15

20

25

t

q1(t

),x 1

(t),

q2(t

),x 2

(t)

x1

q1

q2

x2





Some remarks:





Some remarks:

� The number of steps grows inversely proportional to the square root of thequantization.




Main Features of LIQSS2

Advantages:





Advantages:

� LIQSS2 shares all the advantages of LIQSS.





Advantages:


� LIQSS2 is second-order accurate.





Advantages:



� The number of steps grows with the square root of the accuracy.





Advantages:




� Recent results indicate that LIQSS2 is particularly well suited for simulatingpower electronics circuits.





Advantages:





Disadvantages:





Advantages:





Disadvantages:

� LIQSS2 only looks at the main diagonal of the Jacobian matrix, allowing itsometimes to happen that xj changes its direction without causing amodification of qj . In such cases, LIQSS2 can provoke oscillations depending onthe system structure, i.e., the algorithm may lose its stiffly-stable nature.




LIQSS3: Basic Idea

� LIQSS3 is similar to QSS3. The quantized variables qj (t) are piecewiseparabolic.




LIQSS3: Basic Idea


� Each section of qj (t) starts at xj(t) ± ΔQj with a slope of qj (t) = xj(t)|qj (t)

and a second derivative of qj (t) = xj(t)|qj (t).




LIQSS3: Basic Idea




� We shall attempt to ensure, just as in the cases of LIQSS and LIQSS2, thatxj(t) moves toward qj (t). That is, we demand that

...x j(t) · (qj (t) − xj (t)) > 0.




LIQSS3: Basic Idea





...x j(t) · (qj (t) − xj (t)) > 0.

� When this is not possible, we shall use a linear approximation to find the valueqj (t) with slope qj (t) = xj (t)|qj (t) and with second derivative qj (t) = xj (t)|qj (t),

for which...x j (t) = 0.




LIQSS3: Basic Idea





...x j(t) · (qj (t) − xj (t)) > 0.

� When this is not possible, we shall use a linear approximation to find the valueqj (t) with slope qj (t) = xj (t)|qj (t) and with second derivative qj (t) = xj (t)|qj (t),

for which...x j (t) = 0.

� Although this is an implicit algorithm, there is no need for Newton iterations totake place, as only three possible trajectories can be chosen for qj (t).





Advantages:





Advantages:

� LIQSS3 shares all the advantages of LIQSS2.





Advantages:


� LIQSS3 is third-order accurate.





Advantages:



� The number of steps grows with the cubic root of the accuracy.





Advantages:









Advantages:





Disadvantages:





Advantages:





Disadvantages:

� LIQSS3 only looks at the main diagonal of the Jacobian matrix, allowing itsometimes to happen that

...x j changes its direction without causing a

modification of qj . In such cases, LIQSS3 can provoke oscillations depending onthe system structure, i.e., the algorithm may lose its stiffly-stable nature.


QSS Methods for Marginally Stable Systems

QSS and BQSS and Marginally Stable Systems

Marginally Stable Systems

The following system is known as the harmonic oscillator:

x1(t) = x2(t)

x2(t) = −x1(t)

The harmonic oscillator is the simplest marginally stable system.






x1(t) = x2(t)

x2(t) = −x1(t)


If we consider, for instance, the initial states x1(0) = 4 and x2(0) = 0, we obtain theanalytical solution:

x1(t) = 4 cos(t)

x2(t) = 4 sin(t)






x1(t) = x2(t)

x2(t) = −x1(t)



x1(t) = 4 cos(t)

x2(t) = 4 sin(t)

We already know that F-stable methods are needed to properly integrate such systems.






x1(t) = x2(t)

x2(t) = −x1(t)



x1(t) = 4 cos(t)

x2(t) = 4 sin(t)

We already know that F-stable methods are needed to properly integrate such systems.

Let us check what happen with QSS and BQSS.




QSS, BQSS and Marginally Stable Systems




QSS, BQSS and Marginally Stable Systems II

The results are consistent with what we know about classic time-slicing methods:






� The numerical solution with the forward QSS method gains energy andgenerates output trajectories that are numerically unstable, just as ForwardEuler would.







� The numerical solution with the backward QSS method loses energy andgenerates output trajectories that are asymptotically stable, just as BackwardEuler would.







� The numerical solution with the backward QSS method loses energy andgenerates output trajectories that are asymptotically stable, just as BackwardEuler would.

Evidently, neither of the two methods is suited for the simulation of marginallystable systems.



Centered QSS Method

Centered QSS Method

The simplest F-stable time-slicing method is the trapezoidal rule, which combinesForward and Backward Euler. So, it may make sense to try some combination of QSSand BQSS.



Centered QSS Method

Centered QSS Method


Recall that:



Centered QSS Method

Centered QSS Method


Recall that:

� In QSS, we take qj as the last quantized value intersected by xj .



Centered QSS Method

Centered QSS Method


Recall that:


� In BQSS, we take qj as the next quantized value intersected by xj .



Centered QSS Method

Centered QSS Method


Recall that:


� In BQSS, we take qj as the next quantized value intersected by xj .

Let us see what happens if we take qj as the mean value between the last and thenext quantized values, i.e., if we choose qj in the middle of the quantization interval.We shall call this new method Centered QSS (CQSS) solver.



Centered QSS Method

Centered Quantization

Centered Quantizer

ΔQ

Input

Output



Centered QSS Method

CQSS Simulation of the Harmonic Oscillator



Centered QSS Method

CQSS Features

� CQSS preserves the periodical behavior of the harmonic oscillator.



Centered QSS Method

CQSS Features


� It has been proven that CQSS preserves the geometrical properties of generalHamiltonian (conservative, frictionless) systems: the method is symmetric,symplectic, and it preserves ρ-reversibility.



Centered QSS Method

CQSS Features



� Just like QSS and BQSS, the method does not call for Newton iterations. Infact, it is the only known solver that exhibits F-stability while terminating thecomputations associated with a single step in a fixed amount of real time thatcan be calculated beforehand.



Centered QSS Method

CQSS Features




� Consequently, CQSS is well suited for real-time simulation of marginally stablesystems with low to modest accuracy requirements.



Centered QSS Method

CQSS Features





� Unfortunately, CQSS is only first-order accurate. There are reasons to believethat higher-order CQSS methods cannot be constructed.



Centered QSS Method

CQSS Features





� Unfortunately, CQSS is only first-order accurate. There are reasons to believethat higher-order CQSS methods cannot be constructed.

� It is certainly possible to design higher-order QSS-based solvers that aredecently well suited for the simulation of marginally stable systems, e.g., bycreating a cyclic method, in which steps of QSSi toggle with steps of LIQSSi .Unfortunately, these methods will not preserve the geometric properties ofCQSS in a strict sense, and they will not be truly F-stable.


QSS Methods for Delay Differential Equations

DQSS Methods

Delay Differential Equations

Delay differential equations (DDEs) are similar to ODEs, but their right-handfunctions also depend explicitly on past state values.

x(t) = f(x(t), x(t − τ1(x, t)), ..., x(t − τm(x, t)), u(t))



DQSS Methods




The delay functions τj (·) can be constant, or they can depend on time and/or thestate itself. Instead of an initial state, DDEs need an initial history to be solvable.



DQSS Methods





� DDEs require numerical integration algorithms similar to those for ODEs.



DQSS Methods






� The algorithms are however more complex, as they must look backward in timein order to compute the state derivatives.



DQSS Methods






� The algorithms are however more complex, as they must look backward in timein order to compute the state derivatives.

� From the point of view of the numerical integration, DDEs have some featuresin common with discontinuous ODEs.



DQSS Methods

DDEs and QSS Methods

Recently obtained results show that QSS methods provide an efficient andcomparatively simple solution to the numerical integration of DDEs.



DQSS Methods



Given a DDE:




DQSS Methods



Given a DDE:


The so-called DQSS methods (DQSS1, DQSS2, or DQSS3) approximate it as:

x(t) = f(q(t), q(t − τ1(q, t)), ...,q(t − τm(q, t)), u(t))

where the components of q and x are related by quantization functions of the givenapproximation order.



DQSS Methods

DQSS and PowerDEVS



DQSS Methods

DQSS and PowerDEVS

The block diagram of a DQSScontains:



DQSS Methods

DQSS and PowerDEVS


� hysteretic quantizedintegrators to computeqi (t),



DQSS Methods

DQSS and PowerDEVS



� static functions to computexi (t),



DQSS Methods

DQSS and PowerDEVS




and also:



DQSS Methods

DQSS and PowerDEVS




and also:

� static functions to computeτj (t), and



DQSS Methods

DQSS and PowerDEVS




and also:

� static functions to computeτj (t), and

� a new type of delay blocksthat compute qi (t − τj (t))given qi (t) and τj (t).



DQSS Methods

DQSS and PowerDEVS II

Each of the PowerDEVS blocks can be implemented as a DEVS model.



DQSS Methods



� We already have seen the DEVS models for the hysteretic quantized integrators.



DQSS Methods




� We also have seen the DEVS models for static functions.



DQSS Methods





� We only need to still build a DEVS model for the delay block.



DQSS Methods





� We only need to still build a DEVS model for the delay block.

Dd(t) = y(t − τ(t))

τ(t)

y(t)



DQSS Methods

The Delay Block

The inputs of the delay block y(t) and τ(t) and the output d(t) = y(t − τ(t)) arepiecewise polynomial trajectories. The maximum order of the polynomials depends onthe approximation order of the method.



DQSS Methods

The Delay Block


� The first input y(t) will be represented as a sequence of events at times tyk ,

carrying values y0,k , y1,k , y2,k , so that:

y(t) = y0,k + y1,k · (t − tyk ) + y2,k · (t − ty

k )2; for tyk < t < ty

k+1



DQSS Methods

The Delay Block






k+1

� The second input τ(t) will be represented as a sequence of events at times tτj ,

carrying values τ0,j , τ1,j , τ2,j , so that:

τ(t) = τ0,j + τ1,j · (t − tτj ) + τ2,j · (t − tτ

j )2; for tτj < t < tτ

j+1



DQSS Methods

The Delay Block






k+1

� The second input τ(t) will be represented as a sequence of events at times tτj ,

carrying values τ0,j , τ1,j , τ2,j , so that:

τ(t) = τ0,j + τ1,j · (t − tτj ) + τ2,j · (t − tτ

j )2; for tτj < t < tτ

j+1

� The DEVS delay model computes the corresponding event sequences ofd(t) = y(t − τ(t)).



DQSS Methods

The Delay Algorithm

]

]



DQSS Methods

Theoretical Properties of DQSS

Given a DDE with constant delays of the form:

x(t) = A · x(t) +m∑

i=1

Ai · x(t − τi )



DQSS Methods



x(t) = A · x(t) +m∑

i=1

Ai · x(t − τi )

where the analytical solution x(t) = 0 corresponding to the trivial initial history isasymptotically stable, then:



DQSS Methods



x(t) = A · x(t) +m∑

i=1

Ai · x(t − τi )


� The absolute global error committed by any DQSS method is bounded for anyquanta adopted.



DQSS Methods



x(t) = A · x(t) +m∑

i=1

Ai · x(t − τi )



� The error goes to zero when the quantization goes to zero.



DQSS Methods



x(t) = A · x(t) +m∑

i=1

Ai · x(t − τi )



� The error goes to zero when the quantization goes to zero.

There is a similar property for general nonlinear DDE models with state and timedependent delays. In that case, a stronger condition (input-to-state stability) isrequired for the original system, and the global error boundedness is only guaranteedfor sufficiently small quanta.



DQSS Methods

DQSS Features

� The usage of QSS methods for DDE simulations provides highly efficientalgorithms that often improve on the results obtained by classic time-slicingmethods.



DQSS Methods

DQSS Features


� While conventional methods for ODEs must be almost entirely reformulated todeal with DDEs, QSS methods reduce the problem of the delays to polynomialmanipulations locally managed by a new atomic DEVS block. The staticfunctions and hysteretic quantized integrators remain unchanged.



DQSS Methods

DQSS Features


� While conventional methods for ODEs must be almost entirely reformulated todeal with DDEs, QSS methods reduce the problem of the delays to polynomialmanipulations locally managed by a new atomic DEVS block. The staticfunctions and hysteretic quantized integrators remain unchanged.

� DQSS solvers have strong theoretical properties regarding stability, convergence,and accuracy that guarantee the correctness of the results.


QSS Methods and Real Time Simulation

QSS Methods in Real Time


QSS methods have features that make them particularly suitable for runningsimulations in real time:






� They detect and handle discontinuities without iterations.







� They can integrate many stiff systems in predetermined time, i.e., withoutrequiring Newton iterations.








� If real-time speed cannot be attained, accuracy can be sacrificed by increasingthe quantum. This would only affect accuracy, whereas stability will bepreserved.









� They offer dense output. Thus, the simulation outputs can be communicated atany instant of time.









� They offer dense output. Thus, the simulation outputs can be communicated atany instant of time.

� These algorithms are naturally asynchronous. Hence distributing QSSsimulations across a parallel multi-processor architecture is trivial.



PowerDEVS in Real Time


A version of PowerDEVS runs on a real-time operative system called Linux RTAI,with the following features:






� Simulations can be synchronized with the real-time clock with a precision ofroughly 1μs .







� PowerDEVS-RT offers special modules to detect and process hardwareinterrupts.








� PowerDEVS-RT permits to communicate with input and output hardware portsin a very simple fashion.








� PowerDEVS-RT permits to communicate with input and output hardware portsin a very simple fashion.

� PowerDEVS-RT has been successfully used in the real-time simulation of powerelectronic converters, working at frequencies where most real-time simulationtools fail.



Conclusions

Conclusions

� A broad palette of different QSS-based algorithms has been presented in the lastthree presentations.



Conclusions

Conclusions


� QSS-based algorithms have meanwhile proven themselves to offer an interestingand innovative new approach to numerical ODE, DAE, and DDE solutions formany different problems.



Conclusions

Conclusions



� QSS-based algorithms can be highly competitive when used for applicationswith low to average accuracy requirements. Problems calling for high-orderalgorithms cannot be dealt with effectively using QSS-based solvers.



Conclusions

Conclusions



� QSS-based algorithms can be highly competitive when used for applicationswith low to average accuracy requirements. Problems calling for high-orderalgorithms cannot be dealt with effectively using QSS-based solvers.

� Already QSS4 is rarely more cost-effective than QSS3, because the increasedoverhead eats into the benefit to be reaped from the larger step sizes. AlreadyQSS5 will require iterations in every step, because a fifth-order polynomial willhave to be solved, which can only be done iteratively.



Conclusions

Conclusions II

� QSS-based algorithms are of particular interest for the simulation of systemsexhibiting frequent discontinuities, as state events can be handled much moreefficiently by state-quantization algorithms than by time-slicing algorithms.



Conclusions

Conclusions II


� The benefit of QSS-based solvers becomes even more pronounced when thediscontinuous system to be simulated is furthermore stiff. A typical applicationof stiff discontinuous systems are models of switching power electronic circuits.



Conclusions

Conclusions II



� QSS-based solvers are very promising for the simulation of large-scale models, asthey exploit the sparsity inherent in these models naturally and directly.



Conclusions

Conclusions II



� QSS-based solvers are very promising for the simulation of large-scale models, asthey exploit the sparsity inherent in these models naturally and directly.

� QSS-based solvers perform equally well as classical solvers when dealing withsmall-scale models of systems without discontinuities. Yet in those cases, thereis nothing that the QSS solvers can exploit that would make them moreattractive than the much simpler classical ODE solvers.



Conclusions

Conclusions III

� QSS-based algorithms are of particular interest for real-time simulation.



Conclusions

Conclusions III


� Since QSS solvers are naturally asynchronous, they can be easily distributedover a multi-processor architecture.



Conclusions

Conclusions III



� The communication band-width is minimized, because communication betweentwo processors only takes place at event times. As long as not much ishappening, there is no need to communicate at all. Thus, also thecommunication occurs in a naturally asynchronous fashion.



Conclusions

Conclusions III



� The communication band-width is minimized, because communication betweentwo processors only takes place at event times. As long as not much ishappening, there is no need to communicate at all. Thus, also thecommunication occurs in a naturally asynchronous fashion.

� The communication band-width is furthermore minimized, because only a singlebit needs to be communicated between processors to indicate a state change. Amessage of “1” means that the sending state incremented its level, whereas amessage of “0” means that it decremented its level. As long as the statevariable doesn’t change its level, it doesn’t send an output event through itsoutput channel.



References

References

1. Bergero, F., E. Kofman, and F.E. Cellier (2013), “A Novel ParallelizationTechnique for DEVS Simulation of Continuous and Hybrid Systems,”Simulation, in press.

2. Migoni, G., M. Bortolotto, E. Kofman, and F.E. Cellier (2013), “LinearlyImplicit Quantization–based Integration Methods for Stiff Ordinary DifferentialEquations,” Simulation Modelling Practice and Theory, 35(6), pp.118-136.

3. Migoni, G., E. Kofman, and F.E. Cellier (2012), “Quantization-basedNew Integration Methods for Stiff ODEs,” Simulation, 88(4), pp.387-407.

4. Castro, R., E. Kofman, and F.E. Cellier (2011), “Quantization-basedIntegration Methods for Delay Differential Equations,” Simulation ModellingPractice and Theory, 19(1), pp.314-336.

5. Kofman, E., F.E. Cellier, and G. Migoni (2010), “Continuous SystemSimulation and Control,” in: Discrete Event Simulation and Modeling:Theory and Applications (Model-Based Design), (G.A. Wainer and P.J.Mosterman, eds.), CRC Press, Boca Raton, FL, pp.75-107.

http://dx.doi.org/10.1177/0037549712454931

http://dx.doi.org/10.1177/0037549712454931

http://dx.doi.org/10.1016/j.simpat.2013.03.004



http://dx.doi.org/10.1177/0037549711403645

http://dx.doi.org/10.1177/0037549711403645



http://www.crcnetbase.com/doi/pdfplus/10.1201/b10412-6

http://www.crcnetbase.com/doi/pdfplus/10.1201/b10412-6

http://www.crcnetbase.com/doi/book/10.1201/b10412

http://www.crcnetbase.com/doi/book/10.1201/b10412



References

References II

1. Bergero, Federico (2012), Simulaciones de Sistemas Hıbridos por EventosDiscretos: Tiempo Real y Paralelismo, Ph.D. Dissertation, Faculty of ExactSciences, Engineering, and Land-surveying, National University of Rosario,Argentina.

2. Migoni, Gustavo (2010), Simulacion por Cuantificacion de Sistemas Stiff, Ph.D.Dissertation, Faculty of Exact Sciences, Engineering, and Land-surveying,National University of Rosario, Argentina.

http://www.fceia.unr.edu.ar/~fbergero/phd_thesis.pdf

http://www.fceia.unr.edu.ar/~fbergero/phd_thesis.pdf

http://www.inf.ethz.ch/personal/fcellier/Lect/NSDS/Refs/migoni_phd.pdf

Numerical Simulation of Dynamic Systems XXVII · Numerical Simulation of Dynamic Systems XXVII...

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